Different Strategies for Statistical Compact Modeling MOS-AK Dresden 18.3. Dr. Marat Yakupov (MunEDA) Dr. Michael Pronath (MunEDA) Dr. Elmar Gondro (Infineon Technologies)
Overview Process Manufacturing Q&A Strategies for Statistical Compact Modeling Examples of Modeling process variations 2
Abstract Statistical circuit simulations require model parameters that accurately represent not only the standard deviation of process control monitors (PCMs), but also the correlation structure between them. Usually the standard deviations of PCMs are chosen from specification limits of the technology, where are correlations between PCMs can be measured from sample wafers. The difficult task is then, to find the correlation structure of model parameters, which are widely used as independent and normally distributed random sources, so that correct PCM correlations are achieved by Monte-Carlo simulations. From our view, the correlation structure of model parameters can be combined into 5 different groups: uncorrelated, fully correlated, correlated within the same device type, correlated between different device types or by arbitrary correlation structure. In this presentation we review different strategies for selection of correlation structures of model parameters based on their physical meanings. 3
Process Variation Manufacturing QA Raw Material Incoming Quality Control Wafer Processing Wafer Acceptance Test (WAT) Process Control Monitor (PCM) Monitor environment, layer thickness, Daily, bi-weekly or monthly Electrical & Visual, every wafer, 5 points per wafer Circuit Probing (CP) Electrical, every circuit, measuring specs Assembly Final Test Electrical, every circuit, measuring specs Final Inspection (QC) 4
Process Variation Wafer Acceptance Test Process Control Monitor Usual: Measuring 5 data points per wafer Pass/fail parameters: Used for wafer selection (WAT) Process Control Monitor (PCM) Information parameters: do not lead to wafer reject in case of failure Wafer accepted if pass/fail parameters are inside limits for at least 4 points Example: DISPOSITIONAL PASS/FAIL PARAMETERS Parameter Symbol Min Typ Max Unit NWELL sheet resistance PWELL sheet resistance RSHN 54 60 66 / RSHP 76 90 104 / Wafer map with 5 data points: 1 2 5 4 3 5
Only digital DC metrics. No gm, rds, Cgs, Process Variation Wafer Acceptance Test Process Control Monitor (2) Issues Only few parameters are measured. WAT/PCM does not contain all parameters that are necessary for correct MOS operation in analog circuits. Wafers with few WAT /PCM violations still get accepted. Example: DISPOSITIONAL PASS/FAIL PARAMETERS Parameter Symbol Min Typ Max Unit VTH(tri) long channel VTH (sat), long channel IDSAT short channel Drain-Source Breakdown short channel VTLINL 540 600 660 mv VTSATL 560 640 720 mv IDSATS 340 360 380 ua/um VBDSS 4.1 >4.2 - V 6
Process Monte-Carlo Analysis of Process Control Monitors (PCM) CPK Fast min USL 3 ˆ, ˆ LSL 3 1.5 Slow Lower spec limit (LSL) Fab1 4.5σ Process MC Fab 2 2015 +4.5σ Fab 1 2014 Upper spec limit (USL) Nominal VTHSAT Process MC include variation, drift, difference between Fabs Good for design centering Does not enable yield estimation. 7
Process Monte-Carlo model parameters generation Process MC is used to estimate variability of analog performances Advantage of process MC over process corners (ff,ss, ): Correct correlations between devices and parameters in devices (Corners usually correlate regular MOS, moscaps, I/O, to achieve worst-case speed or power of digital logic) More efficient when many device types are used (MOS, RES, CAP, ). (number of corners grows exponentially with device types) Can be used for design centering (measuring safety distances of specs). (Corners: only pass/fail information) Disadvantage to process corners: Does not prove correct function at process limits. Process drift may cause low yield although process MC shows high yield. Use both corners and process MC for analog sign-off 8
Relation between model parameters and PCM Process Control Monitors MODEL PARAMETERS Tox, Vth0, U0, PROCESS CONTROL MONITORS (PCM) Ion, Ioff, Vth_lin, Vth_sat, 9
MunEDA Statistical Fit Project Node Purpose: Create Statistical Models that reproduce long-term process variation Inputs Netlist or schematic to simulate PCM testbenches with model files Target PCM sigmas Samples of PCMs for correlations/ Correlation file of PCMs Results Set of model parameter sigmas (global MC parameters) and correlations between them if necessary Benefit: More realistic sigmas / correlations of global MC parameters for analog design and verification Safer verification Less pessimism in design better utilization of process technology Ultimate goal of statistical compact modeling is to reproduce electrical performances variations by simulations 10
Statistical Fit Project Node Screenshot of the Statistical Fit Project Node Selection of target sigmas Selection of strategy Selection of Samples file or Correlation file 11
Linear dependence of PCM variation on Model parameter variations Assumption of linear dependence of PCM variation from model parameter variation: P = S. M P variation of PCM S sensitivity of PCMs w.r.t. model parameters M (by simulation) M variation of model parameters Sample PCMs are not linear; transform them to alternative: COX measurements of capacitance TOX oxide thickness RON measurements of on-resisitance GON on-conductance ILEAK leakage current log(ileak) logarithm of ILEAK In order to include second order nonlinearities into the dependence of PCMs on parameter variation, consider, quadratic BPV * * I. Stevanovic, C.C.McAndrew, Quadratic Backward Propagation of Variance for Nonlinear Statistical Circuit Modeling, IEEE on CAD, V.28,No.9, 2009 12
Linear dependence of PCM variation on Model parameter variations The variances of target PCMs are defined by specification: (P) are defined from technology, e.g. (USL-LSL)/9 The correlation of PCMs are measured from the FAB: corr(p) are calculated from wafer Covariance matrix of PCM are constructed from variances and correlations of PCMs: cov(p) = (P). corr(p). (P) 13
HV MOSFET LV MOSFET BIPOLAR CAP Sample structure of correlation matrix of PCMs (Measurement) Typical correlation matrix of PCMs (Long-term data 3 years of meas) DEVICES CAPACITOR Different types BIPOLAR NPN PNP Low Voltage MOSFET NMOS (diff geom) PMOS (diff geom) High Voltage MOSFET NMOS (diff geom) PMOS (diff geom) CAP BIPOLAR LV MOSFET HV MOSFET 14
VTH RON GM Why is the correlation between PCMs so important for analog design? Common source single stage amplifier Low frequency gain: A v =g m. r on There is strong negative correlation between g m and r on : GM RON VTH The MC model shall reproduce these correlation 15
VTH GM Typical problem with modeling of the correlation of PCMs Three typical model parameters for process MC: tox oxide thickness u0 mobility vth0 threshold voltage Consider two PCMs from the same device: GM maximum gm VTH_LIN extrapolated Vth The variance of Gm: 60% dependence on u0 40% dependence on tox The variance of VTH_LIN: 100% dependence on vth0 I ds u0/tox. (V gs -vth0-v ds /2) GM u0/tox VTH_LIN vth0 GM VTH If tox, u0 and vth0 are independent random source, then in simulations there is no correlation between GM and VTH_LIN, what is wrong! 16
Strategies for Statistical Compact Modeling Method Purpose Matrix of PCMs Matrix of Model Parameters Full Correlation Matrix Mathematical approach without physical distinguishes of model parameters Full Covariance Full Covariance Uncorrelated Model Parameters Pay attention on correlation of PCMs, but produce uncorrelated model parameters matrix Full Covariance Only diagonal elements 0 0 Uncorrelated Model Parameters ignore PCM correlation Unknown correlation between PCMs, preliminary estimation of model parameters sigmas Only diagonal elements?? Only diagonal elements 0 0 Block Correlation Structure Known correlation structure of model parameters, take into account correlation between PCMs Full Covariance Block-wise model parameters 0 0 Arbitrary Correlation Structure Know correlation structure of model parameters, but not block wise Full Covariance Arbitrary structure 0 0 17
(1) Method: Full Correlation Matrix (*) Assuming special type of covariance matrix of PCMs positive semi definite Produces full matrix of model parameters Advantage: it reproduces correctly correlations of PCMs Drawback: correlations between model parameters are not physically explainable (1) P=S. M ( N pcm N M ) (2) Cov(P) = S. cov(m). S T (3) Cov(P)= Q.. Q T (4) G=Q. sqrt( ) Cov(P)=G. G T (5) S. H=G (6) Cov(M)= H. H T * Klaus-Willi Pieper, Elmar Gondro An effective method for solving the covariance equation for statistical modeling, 2011, IEEE 18
Example of Full Correlation Matrix Method E.g. two different types of Capacitors: Modelled with two parameters and one correlation factor TOXN TOXP cc_toxn_toxp Correlation matrix of measurements Simulation by Monte-Carlo cc(t_gox_cpgop,c_gox_cngop)=-0.83 cc(t_gox_cpgop,c_gox_cngop)=-0.85 19
(2) Method: Uncorrelated Model Parameters * Input is : cov(p) - covariance matrix of PCMs (1) Calculate cov(p) Number of PCMs: N pcm Number of model parameters: N M Required min N pcm is defined by: YES (N pcm. N pcm + N pcm )/2 N M NO (N pcm. N pcm + N pcm )/2 N M As example for 3 model parameters estimation, this method requires only 2 PCMs and 1 correlation factor Too few PCMs Increase number of measurements Output: Sigmas of model parameters Solve Cov(P) = S. cov(m). S T iteratively by quadratic programming with constraints on cov(m) Exit * I. Stevanovic, X.Li, C.C.McAndrew, K.R.Green, G.Gildenblat Statistical modeling of inter-device correlations with BPV, Solid-State Electronics, 54, pp.796-800, 2010 20
Example of Uncorrelated Model Parameters E.g. two different types of Capacitors: Modeled with three model parameters TOX TOXN TOXP Correlation matrix of measurements Simulation by Monte-Carlo cc(t_gox_cpgop,c_gox_cngop)=-0.83 cc(t_gox_cpgop,c_gox_cngop)=-0.82 21
(3) Method: Uncorrelated Model Parameters ignore PCM correlation Covariance matrix of PCM: cov(p) Number of PCMs: N pcm (1) Calculate cov(p) Number of model parameters: N M Required min N pcm is defined by: YES N pcm N M NO N pcm N M Too few PCMs Increase number of measurements Solve (P)=diag(S. (M). S T ) by quadratic programming with constraints on (M) Exit 22
(4) Method: Block Correlation Structure Inputs are Cov(P) covariance matrix of PCMs Blocks indexes of blocs First block is the global variables, which correlate with all others Strong correlation of MOSFETs through capacitance/oxide thickness (1) Calculate cov(p) YES N pcm N M NO Too few PCMs Increase number of measurements Solve Cov(P) = S. cov(m). S T with constraints by elimination of the elements with 0 correlations Exit 23
(5) Method: Arbitrary Correlation Structure Inputs are Cov(P) covariance matrix of PCMs Indexes of blocks for model parametes to be correlated Strong correlation between different types of devices: LV MOSFET and HV MOSFET (1) Calculate cov(p) YES N pcm N M NO Too few PCMs Increase number of measurements Solve by iterations with constraints Cov(P) = S. cov(m). S T Exit 24
Conclusion Ultimate goal of statistical modeling to reproduce the correlations between measurements of PCMs can be achieved There are several strategies developed to obtain desired correlations of PCMs Mathematically rigorous method of transforming covariance matrix leads to unphysical correlation of model parameters, but gives the best fitting accuracy There are several physical meanings for selection of proper correlation of model parameters Based on the assumptions the complexity of model parameters can be significantly reduced The Statistical Fit Project Node is developed for WiCkeD tools suite this can 25
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